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The principal aim of the present paper is to provide a new lower bound for linear forms in two p-adic elliptic logarithms. We refine a result due to Remond and Urfels. Our improvement lies in the dependence on the height of algebraic coefficients of the linear forms. Our bound is the best possible one concerning the height of the coefficients, where all the relevant constants are explicit. We use the argument that relies on the interpolation method of Laurent, on the variable change introduced by Chudnovsky, and on Faa di Bruno’s formula adapted to matrices whose elements are p-adic elliptic logarithmic functions. The bounds would be useful to determine the set of S-integer points on elliptic curves defined over a number field.

LINEAR FORMS IN TWO ELLIPTIC LOGARITHMS IN THE p-ADIC CASE